Quivers, curves, Kac polynomials and the number of stable Higgs bundles

In the early 80's Kac proved that the number of indecomposable representations

of a given quiver (and a given dimension) over a finite field is a polynomial in the size of the finite field.

Hua later gave an explicit formula for these polynomials and subsequent representation-theoretic or

geometric interpretations for these polynomials were given by Crawley-Boevey, Van den Bergh, Hausel 

and others, leading to a beautiful and still mysterious picture.

The aim of this mini-course is to explain a 'global' analog of some of these results, in which the category

of representations of a quiver gets replaced by the category of coherent sheaves on a smooth projective curve.

As an application, we will give a formula for the number of stable Higgs bundles over such a curve defined 

over a finite field.

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